The stiff Neumann problem: Asymptotic specialty and “kissing” domains
Copyright policy )The stiff Neumann problem: Asymptotic specialty and “kissing” domains
We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain [Formula: see text] which is divided into two subdomains: an annulus [Formula: see text] and a core [Formula: see text]. The density and the stiffness constants are of order [Formula: see text] and [Formula: see text] in [Formula: see text], while they are of order 1 in [Formula: see text]. Here [Formula: see text] is fixed and [Formula: see text] is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as [Formula: see text] for any m. In dimension 2 the case when [Formula: see text] touches the exterior boundary [Formula: see text] and [Formula: see text] gets two cusps at a point [Formula: see text] is included into consideration. The possibility to apply the same asymptotic procedure as in the “smooth” case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as [Formula: see text] for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.
- University of Rome Tor Vergata Italy
- Polytechnic University of Turin Italy
- Institute of Problems of Mechanical Engineering Russian Federation
- Yahoo! United States
- Pushkin Leningrad State University Russian Federation
Variational methods for higher-order elliptic equations, Neumann Laplacian, Asymptotic distributions of eigenvalues in context of PDEs, Neumann Laplacian; asymptotics of eigenvalues and eigenfunctions; stiff Neumann problem; domain with cuspidal point; kissing domains, Mathematics - Analysis of PDEs, asymptotics of eigenvalues and eigenfunctions, FOS: Mathematics, domain with cuspidal point, kissing domains, stiff Neumann problem, Singular perturbations in context of PDEs, Analysis of PDEs (math.AP)
Variational methods for higher-order elliptic equations, Neumann Laplacian, Asymptotic distributions of eigenvalues in context of PDEs, Neumann Laplacian; asymptotics of eigenvalues and eigenfunctions; stiff Neumann problem; domain with cuspidal point; kissing domains, Mathematics - Analysis of PDEs, asymptotics of eigenvalues and eigenfunctions, FOS: Mathematics, domain with cuspidal point, kissing domains, stiff Neumann problem, Singular perturbations in context of PDEs, Analysis of PDEs (math.AP)
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